#7883470

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,29)(30,31);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,28)(21,29)(22,25)(23,27)(26,31)(30,32);(9,10,13,16)(11,14,12,15)(18,25)(19,27)(20,26)(29,30);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,20,23,26)(18,21,25,32)(19,28,27,31)(22,30,24,29);(1,2,5,4)(3,6,8,7)(9,12,13,11)(10,15,16,14)(17,21)(18,26)(19,29)(20,25)(22,28)(23,32)(24,31)(27,30);(1,4,5,2)(3,7,8,6)(17,22)(18,27)(19,25)(20,30)(21,31)(23,24)(26,29)(28,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790666);